Lab 5: Intro to Machine Learning

Practice session

Luisa M. Mimmi —   https://luisamimmi.org/

January 26, 2025

GOAL OF TODAY’S PRACTICE SESSION

  • In this Lab session, we will focus on Machine Learning (ML), as introduced in Lecture 5
  • We will review examples of both supervised and unsupervised ML algorithms
    • Supervised ML algorithms examples
    • Logistic regression
    • 🌳 Random Forest / decision trees 🌲
    • Unsupervised ML algorithms examples
      • K-means Clustering
      • PCA for dimension reduction
    • (optional) PLS-DA for classification, a supervised ML alternative to PCA

🟠 ACKNOWLEDGEMENTS

The examples and datasets in this Lab session follow very closely two sources:

  1. The tutorial on “Principal Component Analysis (PCA) in R” by: Statistics Globe

R ENVIRONMENT SET UP & DATA

Needed R Packages

  • We will use functions from packages base, utils, and stats (pre-installed and pre-loaded)
  • We may also use the packages below (specifying package::function for clarity).
# Load pckgs for this R session

# --- General 
library(here)     # tools find your project's files, based on working directory
here() starts at /Users/luisamimmi/Github/R4stats
library(dplyr)    # A Grammar of Data Manipulation

Attaching package: 'dplyr'
The following objects are masked from 'package:stats':

    filter, lag
The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union
library(skimr)    # Compact and Flexible Summaries of Data
library(magrittr) # A Forward-Pipe Operator for R 
library(readr)    # A Forward-Pipe Operator for R 
library(tidyr)    # Tidy Messy Data

Attaching package: 'tidyr'
The following object is masked from 'package:magrittr':

    extract
library(kableExtra) # Construct Complex Table with 'kable' and Pipe Syntax

Attaching package: 'kableExtra'
The following object is masked from 'package:dplyr':

    group_rows
# ---Plotting & data visualization
library(ggplot2)      # Create Elegant Data Visualisations Using the Grammar of Graphics
library(ggfortify)     # Data Visualization Tools for Statistical Analysis Results
library(scatterplot3d) # 3D Scatter Plot

# --- Statistics
library(MASS)       # Support Functions and Datasets for Venables and Ripley's MASS

Attaching package: 'MASS'
The following object is masked from 'package:dplyr':

    select
library(factoextra) # Extract and Visualize the Results of Multivariate Data Analyses
Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa
library(FactoMineR) # Multivariate Exploratory Data Analysis and Data Mining
library(rstatix)    # Pipe-Friendly Framework for Basic Statistical Tests

Attaching package: 'rstatix'
The following object is masked from 'package:MASS':

    select
The following object is masked from 'package:stats':

    filter
library(car)        # Companion to Applied Regression
Loading required package: carData

Attaching package: 'car'
The following object is masked from 'package:dplyr':

    recode
library(ROCR)       # Visualizing the Performance of Scoring Classifiers

# --- Tidymodels (meta package)
library(rsample)    # General Resampling Infrastructure  
library(broom)      # Convert Statistical Objects into Tidy Tibbles

DATASETS for today


In this tutorial, we will use:

Dataset on Breast Cancer Biopsy

Name: Biopsy Data on Breast Cancer Patients
Documentation: See reference on the data downloaded and conditioned for R here https://cran.r-project.org/web/packages/MASS/MASS.pdf
Sampling details: This breast cancer database was obtained from the University of Wisconsin Hospitals, Madison from Dr. William H. Wolberg. He assessed biopsies of breast tumours for 699 patients up to 15 July 1992; each of nine attributes has been scored on a scale of 1 to 10, and the outcome is also known. The dataset contains the original Wisconsin breast cancer data with 699 observations on 11 variables.

Importing Dataset biopsy

  • The data can be interactively obtained form the MASS R package
# (after loading pckg)
# library(MASS)  

# I can call 
utils::data(biopsy)

biopsy variables with description

Variable Type Description
ID character Sample ID
V1 integer 1 - 10 clump thickness
V2 integer 1 - 10 uniformity of cell size
V3 integer 1 - 10 uniformity of cell shape
V4 integer 1 - 10 marginal adhesion
V5 integer 1 - 10 single epithelial cell size
V6 integer 1 - 10 bare nuclei (16 values are missing)
V7 integer 1 - 10 bland chromatin
V8 integer 1 - 10 normal nucleoli
V9 integer 1 - 10 mitoses
class factor benign or malignant

biopsy variables exploration

The biopsy data contains 699 observations of 9 continuous variables, V1, V2, …, V9.

The dataset also contains a character variable: ID, and a factor variable: class, with two levels (“benign” and “malignant”).

# check variable types
str(biopsy)
'data.frame':   699 obs. of  11 variables:
 $ ID   : chr  "1000025" "1002945" "1015425" "1016277" ...
 $ V1   : int  5 5 3 6 4 8 1 2 2 4 ...
 $ V2   : int  1 4 1 8 1 10 1 1 1 2 ...
 $ V3   : int  1 4 1 8 1 10 1 2 1 1 ...
 $ V4   : int  1 5 1 1 3 8 1 1 1 1 ...
 $ V5   : int  2 7 2 3 2 7 2 2 2 2 ...
 $ V6   : int  1 10 2 4 1 10 10 1 1 1 ...
 $ V7   : int  3 3 3 3 3 9 3 3 1 2 ...
 $ V8   : int  1 2 1 7 1 7 1 1 1 1 ...
 $ V9   : int  1 1 1 1 1 1 1 1 5 1 ...
 $ class: Factor w/ 2 levels "benign","malignant": 1 1 1 1 1 2 1 1 1 1 ...

biopsy missing data

There is one incomplete variable V6 = “bare nuclei” with 16 missing values.

  • remember the package skimr for exploring a dataframe?
# check if vars have missing values
biopsy %>% 
  # select only variables starting with "V"
  skimr::skim(starts_with("V")) %>%
  dplyr::select(skim_variable, 
                n_missing)
# A tibble: 9 × 2
  skim_variable n_missing
  <chr>             <int>
1 V1                    0
2 V2                    0
3 V3                    0
4 V4                    0
5 V5                    0
6 V6                   16
7 V7                    0
8 V8                    0
9 V9                    0

LOGISTIC REGRESSION: 1 EXAMPLE of SUPERVISED ML ALGORITHM

Logistic Regr.: review

  • Logistic regression is used to model a binary response variable, e.g. yes|no, or benign|malignant in the biopsy dataset.

  • Logistic regression is a type of Generalized Linear Model (GLM): is a more flexible version of linear regression that can work also for categorical response variables (e.g. logistic regression) or count data (e.g. poisson regression).

  • GLMs (introduced in 1972) provide a unified framework to accommodate response variables that come from a wide range of distributions:

    • the error distribution (or family) of the response variable can be a normal distribution (continuous response), but also a binomial distribution (binary response), a Poisson distribution (count data), etc.
    • the link function is used to model the relationship between the predictors and the response variable. For example, in logistic regression, the link function is the logit function.

Logistic Regr.: logit function

If we have predictor variables like \(x_{1,i}\), \(x_{2,i}\), …, \(x_{k,i}\) and a binary response variable \(y_i\) (where \(y_i = 0\) or \(y_i = 1\)), we need a “special” function (or link function) able to transform the expected value of the response variable into the outcome we’re trying to predict. So:

If \(p_i\) is the probability that \(y_i = 1\)\[ logit (p_i) = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \cdots + \beta_k x_{k,i} \]

…the logit function is defined as:

\[ logit(p_i) = \ln\left( \frac{p_i}{1-p_i} \right) \]

R fits the coefficients \(\beta_0\), \(\beta_1\), …, \(\beta_k\) to the data using the maximum likelihood estimation method.

Biopsy dataset exploration

  • Biopsied cells of 700 breast cancer tumors, used to determine if the tumors were benign or malignant.
  • This determination was based on 9 characteristics of the cells, ranked from 1(benign) to 10(malignant):
    • 1) Clump Thickness – How the cells aggregate. If monolayered they are benign and if clumped on top of each other they are malignant
    • 2) Uniform Size – All cells of the same type should be the same size.
    • 3) Uniform Shape If cells vary in cell shape they could be malignant
    • 4) Marginal Adhesion – Healthy cells have a strong ability to stick together whereas cancerous cells do not
    • 5) Single Epithelial Size – If epithelial cells are not equal in size, it could be a sign of cancer
    • 6) Bare nuclei – If the nucleus of the cell is not surrounded by cytoplasm, the cell could be malignant
    • 7) Bland Chromatin – If the chromatin’s texture is coarse the cell could be malignant
    • 8) Normal Nucleoli – In a healthy cell the nucleoli is small and hard detect via imagery. Enlarged nucleoli could be a sign of cancer
    • 9) Mitosis – cells that multiply at an uncontrollable rate could be malignant

Biopsy dataset cleaning

  • ID is the patient ID to ensure anonymity (not used in the analysis)
  • Class represents the diagnosis, either benign or malignant, of the patient.
  • Our goal is to understand which variables affect the malignancy of a breast cancer tumor.
    • This will help to predict a diagnosis with the least amount of false positives and false negatives.
# remove rows with missing values
biopsy_clean = na.omit(biopsy)
# rename the columns
colnames(biopsy_clean) <- c("ID","Clump_Thickness", 
                  "Uniform_Size", 
                  "Uniform_Shape",
                  "Marginal_Adhesion", 
                  "Single_Epith_Size", 
                  "Bare_Nuclei",
                  "Bland_Chromatin", 
                  "Normal_Nuclei",
                  "Mitosis", 
                  "Class")
# check the structure of the dataset
paint::paint(biopsy_clean)
data.frame [683, 11]
ID                chr 1000025 1002945 1015425 1016277 10170~
Clump_Thickness   int 5 5 3 6 4 8
Uniform_Size      int 1 4 1 8 1 10
Uniform_Shape     int 1 4 1 8 1 10
Marginal_Adhesion int 1 5 1 1 3 8
Single_Epith_Size int 2 7 2 3 2 7
Bare_Nuclei       int 1 10 2 4 1 10
Bland_Chromatin   int 3 3 3 3 3 9
Normal_Nuclei     int 1 2 1 7 1 7
Mitosis           int 1 1 1 1 1 1
Class             fct benign benign benign benign benign ma~
# remove the ID column
biopsy_clean$ID = NULL

Biopsy sample splitting

In ML, it is good practice to split the data into training and testing sets.

  • We will use the training set to fit the model and the testing set to evaluate the model’s performance.
set.seed(123)

split  <-  sample(nrow(biopsy_clean), 0.8*nrow(biopsy_clean))
biopsy_train  <-  biopsy_clean[split,]
biopsy_test  <-  biopsy_clean[-split,]

Indipendent variables’ visualization

The plot (fig-boxplot) shows the distribution of each variables between the 2 classes of the tumor.

  • All variables behave similarly, with values between 1 and 2 classified as benign and values greater than 2 classified as malignant.

  • The spread for malignancy is much larger than benign patients so we will keep it in mind when testing our data.

ggdf = data.frame("Level" = c(biopsy_train$Clump_Thickness, biopsy_train$Uniform_Size,
                              biopsy_train$Uniform_Shape, biopsy_train$Marginal_Adhesion,
                              biopsy_train$Single_Epith_Size, biopsy_train$Bare_Nuclei,
                              biopsy_train$Bland_Chromatin, biopsy_train$Normal_Nuclei,
                              biopsy_train$Mitosis),
                  "Type" = c("Clump Thickness", "Uniform Size", "Unifrom Shape",
                             "Marginal Adhesion", "Single Epithilial Size", "Bare Nuclei",
                             "Bland Chromatin", "Normal Nuclei", "Mitosis"), 
                  "class" = C(biopsy_train$Class))

# Plot
ggplot(ggdf, aes(x = Level, y = class , colour = class)) + 
  geom_boxplot(fill = NA) +
  scale_color_manual(values = c("#005ca1", "#9b2339")) + 
  geom_jitter(aes(fill = class), alpha = 0.25, shape = 21, width = 0.2) +  
  scale_fill_manual(values = c("#57b7ff", "#e07689")) +  
  facet_wrap(~Type, scales = "free") +  
  theme(plot.title = element_text(size = 13,face="bold", color = "#873c4a"),
        axis.text.x = element_text(size=12,face="italic"), 
        axis.text.y = element_text(size=12,face="italic"),
        legend.position = "none") + 
  labs(title = "Distribution of each explanatory variable by tumor class (benign/malignant) in samples" ) + 
  ylab(label = "") + xlab(label = "")

Indipendent variables’ visualization

Figure 1: Boxplot of the independent variables
- each of the 9 variables is plotted against the class of the tumor
- values are consitently higher & more dispersed for malignant tumors

Logistic Regr.: model fitting

  • We fit a logistic regression model to the biopsy_train data using:
    • the glm function with argument family = binomial to specify the logistic regression model;
    • and with Class ~ . to specify an initial model that uses all the variables as predictors (backward elimination approach).
# Building initial model 
model = stats::glm(Class ~ . , family = binomial, data=biopsy_train)
  • Table 1 shows the model summary, with the coefficient estimate for each predictor.
    • the broom::tidy function converts the model summary into a data frame.
broom::tidy(model) %>% 
  mutate('Sign.lev' = case_when(
    `p.value` < 0.001 ~ "***",
    `p.value` < 0.01 ~ "**",
    `p.value` < 0.05 ~ "*",
    TRUE ~ ""))%>%
  mutate(across(where(is.numeric), ~ round(.x, 4))) %>%
  knitr::kable() 

Logistic Regr.: summary

Table 1: Complete logistic regression model
+ coefficients are in the form of natural logarithm of the odds of the event happening
+ positive estimate indicates an increase in the odds of finding a malignant tumor
term estimate std.error statistic p.value signif. lev.
(Intercept) -9.8386 1.2709 -7.7414 0.0000 ***
Clump_Thickness 0.3935 0.1583 2.4857 0.0129 *
Uniform_Size 0.1624 0.2417 0.6719 0.5017
Uniform_Shape 0.2693 0.2509 1.0735 0.2830
Marginal_Adhesion 0.3163 0.1412 2.2405 0.0251 *
Single_Epith_Size 0.0414 0.2001 0.2066 0.8363
Bare_Nuclei 0.4122 0.1060 3.8877 0.0001 ***
Bland_Chromatin 0.5573 0.2105 2.6472 0.0081 **
Normal_Nuclei 0.1498 0.1275 1.1745 0.2402
Mitosis 0.5065 0.3474 1.4578 0.1449

Logistic Regr.: coefficients’ interpretation

As discussed before, in logistic regression, the coefficients are in the form of the natural logarithm of the odds of the response event happening (i.e. \(Y_i = 1\)):

\[logit(p_i) = \ln\left( \frac{p_i}{1-p_i} \right) = -9.5063 + 0.3935 \times Clump\_Thickness + ... + 0.5065 \times Mitosis\]

However, with some algebraic transformation, the logit function can be inverted to obtain the probability of the response event happening as a function of the predictors:

\[p_i = \frac{1}{1 + e^{-(-9.5063 + 0.3935 \times Clump\_Thickness + ... + 0.5065 \times Mitosis)}}\]

This equation represents the logistic regression model’s best-fit line.

–>

Logistic Regr.: multicollinearity

Let’s check for collinearity using the VIF function from the ‘car’ package.

  • A Variance Inflation Factor \(VIF > 5\) indicates that there could be severe correlation between predictor variables.

  • The VIF values are all less than 5, which indicates that there is no severe correlation between predictor variables in the model.

car::vif(model)
  Clump_Thickness      Uniform_Size     Uniform_Shape Marginal_Adhesion 
         1.251164          2.686457          2.662796          1.120309 
Single_Epith_Size       Bare_Nuclei   Bland_Chromatin     Normal_Nuclei 
         1.496135          1.136493          1.135896          1.315500 
          Mitosis 
         1.070308 

Logistic Regr.: improving the model

  • We can use a statistic called the Akaike Information Criterion (AIC) to compare models.
    • In AIC, a penalty is given for including additional variables.
    • The model with the lowest AIC is considered the best model.
  • Try fitting more parsimonious models by removing variables that are not significant.
# For example let's fit a model without the variable `Uniform_Size`
model2 = glm(Class~ .-Uniform_Size, family = binomial, data=biopsy_train)
 
# Compare the AIC values of the 2 models
tibble(Model = c("model", "model2"), 
       AIC = c(AIC(model), AIC(model2) )) 
# A tibble: 2 × 2
  Model    AIC
  <chr>  <dbl>
1 model  100. 
2 model2  98.5

According to the AIC values, the model2 seems better (AIC is lower).

🖍️🖍️

EQUITABLE EQUATIONS = https://www.youtube.com/watch?v=_yNWzP5HfGw

STATQUEST https://www.youtube.com/watch?v=yIYKR4sgzI8&list=PLblh5JKOoLUKxzEP5HA2d-Li7IJkHfXSe

Logistic Regr.: systematic model selection

  • The MASS package’s function stepAIC enables to perform a systematic model selection (by AIC):
    • The direction argument specifies the direction of the stepwise search.
    • The trace argument (if set to TRUE) prints out all the steps.
  • The best_model has removed these variables:
    • Uniform_Shape
    • Single_Epith_Size
    • Normal_Nuclei
  • The best_model has the lowest AIC value (from 100 to 98.5), despite a higher Residual Deviance than the full model (from 80 to 80.6), albeit by a very slight amount.
# Select the best model based on AIC
best_model <- MASS::stepAIC(model, direction = "both", trace = FALSE)

# Compare the AIC values of full and best model
tibble(Model = c("model", "best_model"), 
       AIC = c(AIC(model), AIC(model2)),
       Deviance = c(deviance(model), deviance(best_model))) %>% kable()
Model AIC Deviance
model 100.00985 80.00985
best_model 98.49005 80.60701

Logistic Regr.: model evaluation on test data

  • We can use the predict function to predict the class of the test data using the best model.
  • Then we use the ROCR package evaluate and visualize the classification
    • the performance function calculates the True Positive Rate (TPR) and False Positive Rate (FPR) for various thresholds. From it we can get:
      • tpr = True Positive Rate (TPR) is the proportion of actual positive cases that are correctly predicted as positive.
      • fpr = False Positive Rate (FPR) is the proportion of actual negative cases that are incorrectly predicted as positive.
      • thresholds = the list of thresholds used to calculate the TPR and FPR.
# Fitted value for the test data 205 samples based on model
pred = stats::predict(best_model, biopsy_test, type = "response")
# Create a prediction object for the ROCR package
ROCRPRed <- ROCR::prediction(predictions = pred, labels = biopsy_test$Class)
# Create a performance object (True Positive Rate and False Positive Rate) for various thresholds.
ROCRPerf <- ROCR::performance(ROCRPRed, "tpr", "fpr")
# Extract TPR and FPR data
fpr <- ROCRPerf@x.values[[1]] # x
tpr <- ROCRPerf@y.values[[1]] # y
# List of thresholds used to calculate the TPR and FPR in the performance object
thresholds <- ROCRPerf@alpha.values[[1]]

🙄🤔 Logistic Regr.: visualization of model performance at varying thresholds

# Create a data frame for ggplot2
perf_data <- data.frame(FPR = fpr, TPR = tpr, Threshold = thresholds)

# Plot using ggplot2
library(ggplot2)
ggplot(perf_data, aes(x = FPR, y = TPR, color = Threshold)) +
  geom_line(linewidth = 1) +
  scale_color_viridis_c(option = "C") +  # Optional: better color scale
  labs(title = "TPR vs FPR",
       x = "False Positive Rate [0, 0.1]",
       y = "True Positive Rate [0.8, 1]",
       color = "Threshold ") +
  coord_cartesian(xlim = c(0, 0.1), ylim = c(0.8, 1)) +
  theme_minimal()

The practical use of this is to identify a threshold where the True Positive Rate (TPR) is acceptably high, without a significant increase in False Positive Rate (FPR).

🙄🤔 Logistic Regr.: visualization of model performance at varying thresholds

Threshold color gradient: YELLOW = the model predicts positive only for very confident cases, PURPLE: the model predicts positive for many cases, even with lower confidence.

Logistic Regr.: confusion matrix

We create a confusion matrix with different acceptance rates so we can see how it affects the true and false positivity rates.

Since our data involves diagnosing malignant tumors, it is important to keep the false negative rate low as this would be telling someone who has a malignant tumor that it is benign.

  • We found that a cutoff of 0.4 gives a good balance of low false negatives while still maintaining a high true positive rate.
table(ActualValue=biopsy_test$Class, PredictedValue=pred>0.4)
           PredictedValue
ActualValue FALSE TRUE
  benign       88    2
  malignant     3   44
table(ActualValue=biopsy_test$Class, PredictedValue=pred>0.7)
           PredictedValue
ActualValue FALSE TRUE
  benign       88    2
  malignant     6   41

Logistic Regr.: ROC curve

predicted.test <- predict(best_model, biopsy_test, type = "response")

predicted.data <- data.frame(prob.of.malig=predicted.test, malig = biopsy_test$Class)

predicted.data <- predicted.data[order(predicted.data$prob.of.malig, decreasing = F),]

predicted.data$rank <- 1:nrow(predicted.data)

plot_ROC <- ggplot(data=predicted.data, aes(x=rank, y=prob.of.malig)) +
  geom_point(aes(color=malig), alpha=1, shape=4, stroke=2) +
  xlab("Index") + 
  ylab("Predicted Probability of Tumor Being Malignant")  

plot_ROC

Logistic Regr.: ROC curve

This shows why lowering the cutoff improves the accuracy of the model as some malignant tumors are being underestimated which would cause false negatives.

Logistic Regr.: conclusions

  • Uniform Size and Single Epithithial Size were not significant in predicting the malignancy of tumor cells so our model does not include these variables.

  • Our fitted model reduces the null deviance and AIC without impacting the residual deviance by a significant amount and is able to predict the testing dataset with >90% accuracy.

  • For further analysis, we could run the model multiple times because our original and revised model are similar.

  • New training and testing data would help confirm our results and help identify possible overfitting.

🟠 K-MEANS CLUSTERING: EXAMPLE of UNSUPERVISED ML ALGORITHM

PCA: EXAMPLE of UNSUPERVISED ML ALGORITHM

Reducing high-dimensional data to a lower number of variables

biopsy dataset manipulation

We will:

  • exclude the non-numerical variables (ID and class) before conducting the PCA.

  • exclude the individuals with missing values using the na.omit() or filter(complete.cases() functions.

  • We can do both in 2 equivalent ways:


with base R (more compact)

# new (manipulated) dataset 
data_biopsy <- na.omit(biopsy[,-c(1,11)])

with dplyr (more explicit)

# new (manipulated) dataset 
data_biopsy <- biopsy %>% 
  # drop incomplete & non-integer columns
  dplyr::select(-ID, -class) %>% 
  # drop incomplete observations (rows)
  dplyr::filter(complete.cases(.))

biopsy dataset manipulation

We obtained a new dataset with 9 variables and 683 observations (instead of the original 699).

# check reduced dataset 
str(data_biopsy)
'data.frame':   683 obs. of  9 variables:
 $ V1: int  5 5 3 6 4 8 1 2 2 4 ...
 $ V2: int  1 4 1 8 1 10 1 1 1 2 ...
 $ V3: int  1 4 1 8 1 10 1 2 1 1 ...
 $ V4: int  1 5 1 1 3 8 1 1 1 1 ...
 $ V5: int  2 7 2 3 2 7 2 2 2 2 ...
 $ V6: int  1 10 2 4 1 10 10 1 1 1 ...
 $ V7: int  3 3 3 3 3 9 3 3 1 2 ...
 $ V8: int  1 2 1 7 1 7 1 1 1 1 ...
 $ V9: int  1 1 1 1 1 1 1 1 5 1 ...

Calculate Principal Components

The first step of PCA is to calculate the principal components. To accomplish this, we use the prcomp() function from the stats package.

  • With argument “scale = TRUE” each variable in the biopsy data is scaled to have a mean of 0 and a standard deviation of 1 before calculating the principal components (just like option Autoscaling in MetaboAnalyst)
# calculate principal component
biopsy_pca <- prcomp(data_biopsy, 
                     # standardize variables
                     scale = TRUE)

Analyze Principal Components

Let’s check out the elements of our obtained biopsy_pca object

  • (All accessible via the $ operator)
names(biopsy_pca)
[1] "sdev"     "rotation" "center"   "scale"    "x"       

“sdev” = the standard deviation of the principal components

“sdev”^2 = the variance of the principal components (eigenvalues of the covariance/correlation matrix)

“rotation” = the matrix of variable loadings (i.e., a matrix whose columns contain the eigenvectors).

“center” and “scale” = the means and standard deviations of the original variables before the transformation;

“x” = the principal component scores (after PCA the observations are expressed in principal component scores)

Analyze Principal Components (cont.)

We can see the summary of the analysis using the summary() function

  1. The first row gives the Standard deviation of each component, which can also be retrieved via biopsy_pca$sdev.
  2. The second row shows the Proportion of Variance, i.e. the percentage of explained variance.
summary(biopsy_pca)
Importance of components:
                          PC1     PC2     PC3     PC4     PC5     PC6     PC7
Standard deviation     2.4289 0.88088 0.73434 0.67796 0.61667 0.54943 0.54259
Proportion of Variance 0.6555 0.08622 0.05992 0.05107 0.04225 0.03354 0.03271
Cumulative Proportion  0.6555 0.74172 0.80163 0.85270 0.89496 0.92850 0.96121
                           PC8     PC9
Standard deviation     0.51062 0.29729
Proportion of Variance 0.02897 0.00982
Cumulative Proportion  0.99018 1.00000

Proportion of Variance for components

  1. The row with Proportion of Variance can be either accessed from summary or calculated as follows:
# a) Extracting Proportion of Variance from summary
summary(biopsy_pca)$importance[2,]
    PC1     PC2     PC3     PC4     PC5     PC6     PC7     PC8     PC9 
0.65550 0.08622 0.05992 0.05107 0.04225 0.03354 0.03271 0.02897 0.00982 
# b) (same thing)
round(biopsy_pca$sdev^2 / sum(biopsy_pca$sdev^2), digits = 5)
[1] 0.65550 0.08622 0.05992 0.05107 0.04225 0.03354 0.03271 0.02897 0.00982


The output suggests the 1st principal component explains around 65% of the total variance, the 2nd principal component explains about 9% of the variance, and this goes on with diminishing proportion for each component.

Cumulative Proportion of variance for components

  1. The last row from the summary(biopsy_pca), shows the Cumulative Proportion of variance, which calculates the cumulative sum of the Proportion of Variance.
# Extracting Cumulative Proportion from summary
summary(biopsy_pca)$importance[3,]
    PC1     PC2     PC3     PC4     PC5     PC6     PC7     PC8     PC9 
0.65550 0.74172 0.80163 0.85270 0.89496 0.92850 0.96121 0.99018 1.00000 


Once you computed the PCA in R you must decide the number of components to retain based on the obtained results.

VISUALIZING PCA OUTPUTS

Scree plot

There are several ways to decide on the number of components to retain.

  • One helpful option is visualizing the percentage of explained variance per principal component via a scree plot.
    • Plotting with the fviz_eig() function from the factoextra package
# Scree plot shows the variance of each principal component 
factoextra::fviz_eig(biopsy_pca, 
                     addlabels = TRUE, 
                     ylim = c(0, 70))


Visualization is essential in the interpretation of PCA results. Based on the number of retained principal components, which is usually the first few, the observations expressed in component scores can be plotted in several ways.

Scree plot

The obtained scree plot simply visualizes the output of summary(biopsy_pca).

Principal Component Scores

After a PCA, the observations are expressed as principal component scores.

  1. We can retrieve the principal component scores for each Variable by calling biopsy_pca$x, and store them in a new dataframe PC_scores.
  2. Next we draw a scatterplot of the observations – expressed in terms of principal components
# Create new object with PC_scores
PC_scores <- as.data.frame(biopsy_pca$x)
head(PC_scores)

It is also important to visualize the observations along the new axes (principal components) to interpret the relations in the dataset:

Principal Component Scores

        PC1         PC2         PC3         PC4         PC5         PC6
1  1.469095 -0.10419679  0.56527102  0.03193593 -0.15088743 -0.05997679
2 -1.440990 -0.56972390 -0.23642767  0.47779958  1.64188188  0.48268150
3  1.591311 -0.07606412 -0.04882192  0.09232038 -0.05969539  0.27916615
4 -1.478728 -0.52806481  0.60260642 -1.40979365 -0.56032669 -0.06298211
5  1.343877 -0.09065261 -0.02997533  0.33803588 -0.10874960 -0.43105416
6 -5.010654 -1.53379305 -0.46067165 -0.29517264  0.39155544 -0.11527442
         PC7        PC8          PC9
1 -0.3491471 -0.4200360 -0.005687222
2  1.1150819 -0.3792992  0.023409926
3 -0.2325697 -0.2096465  0.013361828
4  0.2109599  1.6059184  0.182642900
5 -0.2596714 -0.4463277 -0.038791241
6 -0.3842529  0.1489917 -0.042953075

Principal Component Scores plot (adding label variable)

  1. When data includes a factor variable, like in our case, it may be interesting to show the grouping on the plot as well.
  • In such cases, the label variable class can be added to the PC set as follows.
# retrieve class variable
biopsy_no_na <- na.omit(biopsy)
# adding class grouping variable to PC_scores
PC_scores$Label <- biopsy_no_na$class


The visualization of the observation points (point cloud) could be in 2D or 3D.

Principal Component Scores plot (2D)

The Scores Plot can be visualized via the ggplot2 package.

  • grouping is indicated by argument the color = Label;
  • geom_point() is used for the point cloud.
ggplot(PC_scores, 
       aes(x = PC1, 
           y = PC2, 
           color = Label)) +
  geom_point() +
  scale_color_manual(values=c("#245048", "#CC0066")) +
  ggtitle("Figure 1: Scores Plot") +
  theme_bw()

Principal Component Scores plot (2D)

Figure 1 shows the observations projected into the new data space made up of principal components

Principal Component Scores (2D Ellipse Plot)

Confidence ellipses can also be added to a grouped scatter plot visualized after a PCA. We use the ggplot2 package.

  • grouping is indicated by argument the color = Label;
  • geom_point() is used for the point cloud;
  • the stat_ellipse() function is called to add the ellipses per biopsy group.
ggplot(PC_scores, 
       aes(x = PC1, 
           y = PC2, 
           color = Label)) +
  geom_point() +
  scale_color_manual(values=c("#245048", "#CC0066")) +
  stat_ellipse() + 
  ggtitle("Figure 2: Ellipse Plot") +
  theme_bw()

Principal Component Scores (2D Ellipse Plot)

Figure 2 shows the observations projected into the new data space made up of principal components, with 95% confidence regions displayed.

Principal Component Scores plot (3D)

A 3D scatterplot of observations shows the first 3 principal components’ scores.

  • For this one, we need the scatterplot3d() function of the scatterplot3d package;
  • The color argument assigned to the Label variable;
  • To add a legend, we use the legend() function and specify its coordinates via the xyz.convert() function.
# 3D scatterplot ...
plot_3d <- with(PC_scores, 
                scatterplot3d::scatterplot3d(PC_scores$PC1, 
                                             PC_scores$PC2, 
                                             PC_scores$PC3, 
                                             color = as.numeric(Label), 
                                             pch = 19, 
                                             main ="Figure 3: 3D Scatter Plot", 
                                             xlab="PC1",
                                             ylab="PC2",
                                             zlab="PC3"))

# ... + legend
legend(plot_3d$xyz.convert(0.5, 0.7, 0.5), 
       pch = 19, 
       yjust=-0.6,
       xjust=-0.9,
       legend = levels(PC_scores$Label), 
       col = seq_along(levels(PC_scores$Label)))

Principal Component Scores plot (3D)

Figure 3 shows the observations projected into the new 3D data space made up of principal components.

Biplot: principal components v. original variables

Next, we create another special type of scatterplot (a biplot) to understand the relationship between the principal components and the original variables.
In the biplot each of the observations is projected onto a scatterplot that uses the first and second principal components as the axes.

  • For this plot, we use the fviz_pca_biplot() function from the factoextra package
    • We will specify the color for the variables, or rather, for the “loading vectors”
    • The habillage argument allows to highlight with color the grouping by class
factoextra::fviz_pca_biplot(biopsy_pca, 
                repel = TRUE,
                col.var = "black",
                habillage = biopsy_no_na$class,
                title = "Figure 4: Biplot", geom="point")

Biplot: principal components v. original variables

The axes show the principal component scores, and the vectors are the loading vectors

Interpreting biplot output

Biplots have two key elements: scores (the 2 axes) and loadings (the vectors). As in the scores plot, each point represents an observation projected in the space of principal components where:

  • Biopsies of the same class are located closer to each other, which indicates that they have similar scores referred to the 2 main principal components;
  • The loading vectors show strength and direction of association of original variables with new PC variables.

As expected from PCA, the single PC1 accounts for variance in almost all original variables, while V9 has the major projection along PC2.

Interpreting biplot output (cont.)

scores <- biopsy_pca$x

loadings <- biopsy_pca$rotation
# excerpt of first 2 components
loadings[ ,1:2] 
          PC1         PC2
V1 -0.3020626 -0.14080053
V2 -0.3807930 -0.04664031
V3 -0.3775825 -0.08242247
V4 -0.3327236 -0.05209438
V5 -0.3362340  0.16440439
V6 -0.3350675 -0.26126062
V7 -0.3457474 -0.22807676
V8 -0.3355914  0.03396582
V9 -0.2302064  0.90555729

Recap of the workshop’s content

TOPICS WE COVERED

  1. Motivated the choice of learning/using R for scientific quantitative analysis, and lay out some fundamental concepts in biostatistics with concrete R coding examples.

  2. Consolidated understanding of inferential statistic, through R coding examples conducted on real biostatistics research data.

  3. Discussed the relationship between any two variables, and introduce a widely used analytical tool: regression.

  4. Presented a popular ML technique for dimensionality reduction (PCA), performed both with MetaboAnalyst and R.

  5. Introduction to power analysis to define the correct sample size for hypotheses testing and discussion of how ML approaches deal with available data.

Final thoughts

  • While the workshop only allowed for a synthetic overview of fundamental ideas, it hopefully provided a solid foundation on the most common statistical analysis you will likely run in your daily work:

    • Thorough understanding of the input data and the data collection process
    • Univariate and bivariate exploratory analysis (accompanied by visual intuition) to form hypothesis
    • Upon verifying the assumptions, we fit data to hypothesized model(s)
    • Assessment of the model performance (\(R^2\), \(Adj. R^2\), \(F-Statistic\), etc.)
  • You should now have a solid grasp on the R language to keep using and exploring the huge potential of this programming ecosystem

  • We only scratched the surface in terms of ML classification and prediction models, but we got a hang of the fundamental steps and some useful tools that might serve us also in more advanced analysis